Simulation of Limit Order Books

Most electronic stock exchanges nowadays organize trades via limit order books (LOBs), in order to facilitate trading activity. A limit order book is a centralized record of all outstanding buy (sell) limit orders, which indicate a buyer's (seller's) offer to buy (sell) a specified quantity of a particular stock for a certain price, at a given venue (XETRA, NASDAQ, etc.). These orders remain in the order book until they get either cancelled or executed by a sell (buy) market order.

Due to the availability of vasts amounts of historical limit order book data, a lot of research has been devoted in recent years to study empirical aspects of limit order books, investigate how the price is formed within limit order books, how price movements in limit order books may be predicted, as well as methods to simulate limit order books. All those tasks are of complicated nature as many heterogeneous agents with different objectives and different sets of information interact together in limit order books. This turns the limit order book of a stock, future, etc. into a complex, high-dimensional system evolving over time with non-trivial dynamics.

The aim of this research project is the development of models to simulate limit order book data which reproduces certain empirical properties observed in real limit order books. These properties regard the resulting price paths, order book shape and order inter-arrival times, as well as non-stationarity patterns. Ultimately, limit order books are responsive systems in which actions affect each other. In essence, if some trader places a limit order, the market (limit order book) reacts to this placement which changes the development of the LOB. Thus, a "useful" model should also be able to simulate the reactions of a limit order book realistically, if an order is placed in the book. Models to simulate LOBs are useful for a variety of things. First of all, they help to better understand financial markets in a variety of scenarios (e.g. during crises). Second, it alleviates issues regarding data sharing. Third, enrichment of data sets from tail events (crises, etc.) can help render financial algorithms more robust, as they can be tested in more critical situations than what is available in historical data.

Point processes, multi-agent systems and more have been applied to model limit order books, and are able to reproduce some of the typical characteristics observed in limit order books, but most approaches lack in certain dimensions. Recently, the advent of machine learning and advances in computational power allowed to apply data-driven, non-parametric approaches for the simulation of financial data on a much larger scale than ever before. Despite their lack of analytical tractability, machine learning algorithms tend to be more flexible than conventional models. In particular, generative adversarial networks (GANs) and their extensions have recently been applied to (multivariate) time series generation. The research applying generative adversarial networks to generate limit order book data is yet very limited to. This project aims to develop novel GAN architectures suitable to generate limit order book data, e.g. processes of quantities and prices, with realistic properties which are lacking from the current literature.

The project is aligned with the following topics: (1) Artificial intelligence technologies, (2) statistics and applied probability, (3) non-linear systems, (4) operational research.
The industrial partner/collaborator is J.P.Morgan.

Probabilistic modelling permeates the Financial services, healthcare, technology and other Service industries crucial to the UK's continuing social and economic prosperity, which are major users of stochastic algorithms for data analysis, simulation, systems design and optimisation. There is a major and growing skills shortage of experts in this area, and the success of the UK in addressing this shortage in cross-disciplinary research and industry expertise in computing, analytics and finance will directly impact the international competitiveness of UK companies and the quality of services delivered by government institutions.
By training highly skilled experts equipped to build, analyse and deploy probabilistic models, the CDT in Mathematics of Random Systems will contribute to
- sharpening the UK's research lead in this area and
- meeting the needs of industry across the technology, finance, government and healthcare sectors

MATHEMATICS, THEORETICAL PHYSICS and MATHEMATICAL BIOLOGY

The explosion of novel research areas in stochastic analysis requires the training of young researchers capable of facing the new scientific challenges and maintaining the UK's lead in this area. The partners are at the forefront of many recent developments and ideally positioned to successfully train the next generation of UK scientists for tackling these exciting challenges.
The theory of regularity structures, pioneered by Hairer (Imperial), has generated a ground-breaking approach to singular stochastic partial differential equations (SPDEs) and opened the way to solve longstanding problems in physics of random interface growth and quantum field theory, spearheaded by Hairer's group at Imperial. The theory of rough paths, initiated by TJ Lyons (Oxford), is undergoing a renewal spurred by applications in Data Science and systems control, led by the Oxford group in conjunction with Cass (Imperial). Pathwise methods and infinite dimensional methods in stochastic analysis with applications to robust modelling in finance and control have been developed by both groups.
Applications of probabilistic modelling in population genetics, mathematical ecology and precision healthcare, are active areas in which our groups have recognized expertise.

FINANCIAL SERVICES and GOVERNMENT

The large-scale computerisation of financial markets and retail finance and the advent of massive financial data sets are radically changing the landscape of financial services, requiring new profiles of experts with strong analytical and computing skills as well as familiarity with Big Data analysis and data-driven modelling, not matched by current MSc and PhD programs. Financial regulators (Bank of England, FCA, ECB) are investing in analytics and modelling to face this challenge. We will develop a novel training and research agenda adapted to these needs by leveraging the considerable expertise of our teams in quantitative modelling in finance and our extensive experience in partnerships with the financial institutions and regulators.

DATA SCIENCE:

Probabilistic algorithms, such as Stochastic gradient descent and Monte Carlo Tree Search, underlie the impressive achievements of Deep Learning methods. Stochastic control provides the theoretical framework for understanding and designing Reinforcement Learning algorithms. Deeper understanding of these algorithms can pave the way to designing improved algorithms with higher predictability and 'explainable' results, crucial for applications.
We will train experts who can blend a deeper understanding of algorithms with knowledge of the application at hand to go beyond pure data analysis and develop data-driven models and decision aid tools
There is a high demand for such expertise in technology, healthcare and finance sectors and great enthusiasm from our industry partners. Knowledge transfer will be enhanced through internships, co-funded studentships and paths to entrepreneurs